Accurate Theoretical Description of the 1La and 1Lb Excited States in

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An Accurate Theoretical Description of the 1La and 1Lb Excited States in Acenes using the all Order Constricted Variational Density Functional Theory Method and the Local Density Approximation. Mykhaylo Krykunov, Stefan Grimme, and Tom Ziegler J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/ct300372x • Publication Date (Web): 23 Aug 2012 Downloaded from http://pubs.acs.org on September 8, 2012

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Journal of Chemical Theory and Computation

An Accurate Theoretical Description of the 1La and 1Lb Excited States in Acenes using the all Order Constricted Variational Density Functional Theory Method and the Local Density Approximation. Mykhaylo Krykunov,*,a Stefan Grimme, b and Tom Ziegler a Department of Chemistry, University of Calgary, University Drive 2500, Calgary, Alberta , Canada T2N 1N4. Mulliken Center for Theoretical Chemistry, Institut für Physikalische und Theoretische Chemie der Universität Bonn, Beringstr. 4,D-‐53115, Germany Abstract: We present the results of calculations on the vertical singlet 1La and 1Lb excitation energies in acenes within time dependent density functional theory (TDDFT), second order constricted variational DFT (CV(2)-‐DFT) and all order constricted variational DFT (CV(∞)-‐DFT) using the local density approximation LDA(VWN). For the linear acenes it is shown that the application of the Tamm-‐ Dancoff (TD) approximation to TDDFT (TDDFT-‐TD) substantially improves the agreement with experiment compared to pure TDDFT. This improvement leads to the correct ordering of the 1La and 1Lb excitation energies in Naphthalene. As TDDFT-‐TD is equivalent to the second order CV(2)-‐TD method one might hope for further improvements by going to all orders in CV(∞)-‐TD. Indeed, for linear acenes the application of the CV(∞)-‐TD method brings the agreement with experiment to within 0.1 eV for both types of excitations using the simple LDA functional. The CV(∞)-‐TD method based on LDA is also shown to be accurate for 15 nonlinear acenes with root mean square deviations of 0.24 eV for 1La and 0.17 eV for 1Lb. 1 Introduction A number of π-‐conjugated systems in the form of organic dyes have recently attracted considerable attention as possible candidates that might be incorporated into dye-‐sensitized solar cells. To devise the optimal cell, it would be very helpful to have an accurate theoretical method that could search for good candidates among * Corresponding author: E-‐mail [email protected] aUniversity of Calgary b University of Bonn

ACS Paragon Plus Environment

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the many possible dyes in a both efficient and accurate manner. In order to establish such a method, it is also necessary to have a set of benchmarks among π → π∗ excitation energies in extended π-‐conjugated systems. Fortunately, the singlet 1La and 1Lb excitations in linear poly-‐acenes represent such a set of benchmark data as they have been studied extensively both experimentally and theoretically. Here the acenes consist of a number (nr) of fused benzene rings. The distinct properties1 of the 1La and 1Lb states for the linear acenes have already been described in the literature.2-‐3 Essentially the 1La (or 1B2u when the x-‐ axis corresponds to the long molecular axis ) state is dominated by a single electron transition HOMO → LUMO , while the 1Lb (or 1B3u) state results from a combination of HOMO-‐1 → LUMO and HOMO → LUMO+1 transitions. Further, excitations to 1La (1B2u) are short axis polarized with high intensity whereas the transitions to 1Lb (1B3u) are long axis polarized with low intensity, Table 1. The relative ordering of the 1La and 1Lb states is size dependent. Thus, for naphthalene with two rings the 1Lb state is below 1La in energy while for the other linear acenes the ordering is opposite although the splitting is very small for anthracene with nr =3. Both states drop in energy with increasing nr but the drop is largest for 1La , see Table 1 and Figure 1. It was previously reported2,3 that TDDFT4-‐7 incorrectly predicts the crossing point of the 1La and 1Lb states in linear poly-‐acenes with both semi-‐local and hybrid functionals. Further, the computed TDDFT excitation energies for 1La relative to 1Lb are in poor agreement with experiment throughout the range nr =2,6.8-‐9 Promising alternative methods are the DFT/MRCI approach11-‐12 and TDDFT schemes based on


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long-‐range corrected (LRC) functionals.13-‐16 The DFT/MRCI approach is quite accurate but not a “black box” method like the TDDFT scheme. This makes it difficult to use DFT/MRCI in screening type applications. The LRC methods show some improvement for 1La but not for 1Lb.16 Further, the optimal range separation parameter(s) must be determined for each molecule. A more recent promising approach is double-‐hybrid density functional theory.20-‐25 This scheme has to date provided the most balanced description of the linear acenes.25 Unfortunately, these double hybrids are limited to low valence excitations due to the inherent perturbative corrections.26 Some of us have recently put forward27 the constricted variational density functional theory (CV-‐DFT) as a possible improvement of adiabatic TDDFT. The new theory is in its simplest second order formulation (CV(2)-‐DFT) identical to adiabatic TDDFT after use has been made of the popular Tamm-‐Dancoff approximation.28 However the method can be extended to all orders in the variational parameters (CV(∞)-‐DFT). It has been shown in a recent study27 that the π → π∗ transitions in conjugated hydrocarbons are described reasonably well within the CV(∞)-‐DFT scheme. An average deviation from experiment is approximately 0.3 eV for both semi-‐local and hybrid functionals. We shall here first assess the ability of our newly developed constricted variational density functional theory (CV-‐DFT)27 to describe the excited states in linear poly-‐acenes, as a first step in an attempt to establish an inexpensive method that ultimately can help in the design of light-‐harvesting devices. It is further our


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objective to demonstrate that quite accurate results can be obtained for our test set even with the simple LDA functional if one goes beyond the simple second order approximation inherent in TDDFT and incorporate terms to all orders in the variational parameters, as it is done in CV(∞)-‐DFT. Our work is a continuation of a previous study in which we demonstrated that LDA can be used to describe charge transfer transitions within the constricted variational DFT method. 29 We shall finally extend the scope of our investigation to include the 1La and 1Lb excitations in 15 nonlinear poly-‐acenes. 2 Computational Method and Computational Details 2.1 Constricted Variational Density Functional theory

In CV(n)-‐DFT, we carry out a unitary transformation among occupied

{" i ;i = 1, occ} and virtual {" a ; a = 1, vir } ground state orbitals

!

!!! occ Y # # !vir "

$ ! ! $ ! m=' m $! ! $ ! !' U occ occ U &=e # &=# ( & = # occ &# & & # !vir & # # & # ' % " % " m=0 m! %" !vir % #" !vir

$ & && (1) %

with the summation over m in (1) truncated at m=n. Thus the resulting orbitals will be orthonormal to order n in U . Here " oc c and " vir are concatenated column vectors consisting of the sets {" i ;i = 1, occ} and {" a ; a = 1, vir } of occupied and virtual ground

! !" ' and " ' are concatenated column vectors state orbitals, respectively, whereas oc c vir ! ! ' ;i = 1, occ} and {" ' ;a = 1, vir} of occupied and virtual containing the resulting sets {" i a

! ! unitary transformation matrix Y is in eq 1 excited state orbitals, respectively. The ! matrix U as expressed in terms o!f a skew symmetric

!

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! ! U2 U m ! (U 2 )m (U 2 )m Y = e = I +U + (2) +! = " =" +U" 2 m=0 m! m=0 2m! m=0 (2m +1)! U

Here Uij = U ab = 0 where “i,j” refer to the occupied set {" i ; i = 1, occ } whereas “a,b” refer to {" a ; a = 1, vir } . Further, U ai are the variational mixing matrix elements that

! combine virtual and occupied ground state !orbitals in the excited state with

!U ai = !Uia . The new occupied ! set {" 'i ;i = 1, occ} provides us with the excited state KS-‐ determinant ! ' = !1'!2' ...!i'! 'j ...!n' from which we can express the corresponding spin

! and space density matrices s' and ! ' . These matrices allow us subsequently to express the Kohn-‐Sham energy of ! ' as E[ ! '+, ! '! ] where ! '+ = ! ' / 2 + s ' / 2 and

! '! = ! ' / 2 ! s' / 2 .27 In CV(2)-‐DFT the summation in eq 1 is carried out to m=2 so that the corresponding orbitals are orthonormal up to second order in U. From that

E[ ! '+, ! '! ] is expressed to second order in U from the second order density and spin matrices as E[ ! (2)+, ! (2)! ] . A subsequent optimization of E[ ! (2)+, ! (2)! ] under the constraint27 that exactly one electron is moved from the density space spanned by

{" i ; i = 1, occ } to the space spanned by {" a ; a = 1, vir } affords after applying the Tamm-‐Dancoff approximation28 the eigenvalue equation

!

! AU(I ) = ! (I )U (I ) (3) where Aai,bj = "ab"ij (# a $ # i ) + K ai,bj

(4).

!


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We shall refer to the CV(2)-DFT method in which we have used the Tamm-‐Dancoff approximation28 as simply CV(2)-‐TD. C XC In eq 4 ! a and !i refer to ground state orbital energies. Further K ai,bj = K ai,bj + K ai,bj

where

K

C ai,bj

=

!

! 1 ! !a (1)!i (1) r !b (2)! j (2)dv1dv2 (5) 12

whereas XC(HF ) K ai,bj =!"

1

" ! (1)! (2) r a

i

!b (1)! j (2)dv1dv2 (6)

12

for Hartree Fock exchange correlation and XC(KS ) K ai,bj =

! ! ! (1)! (1) f (1)! (1)! (1)dv dv a

i

b

j

1

2

(7)

for KS exchange correlation. In eq 7 f is the energy kernel. We note that an expression similar to eq 3 can be derived4 from adiabatic TDDFT after application of the Tamm-‐ Dancoff approximation.28 Thus adiabatic TDDFT-‐TD based on response theory and CV(2)-‐TD based on variation theory are mathematically equivalent.

After solving eq 3 we apply the resulting U to eqs 1 and 2 in an all order

unitary transformation from which we can obtain E[ ! (!)+, ! (!)" ] corresponding to CV(∞)-‐DFT. In general, in CV(n)-‐DFT we use the U from eq 3 to express

E[ ! (n)+, ! (n)! ] . It is also possible to determine U directly by optimizing E[ ! (n)+, ! (n)! ] . In that case we talk about SCF-‐CV(n)-‐DFT. 27 We shall here only be interested in CV(n)-‐DFT as we want the simplest possible theory that can describe π → π∗


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excitation in dyes. However, the π → π∗ excitations in conjugated hydrocarbons have been studied extensively by SCF-‐CV(n)-‐DFT. 27

We shall now turn to a discussion of how we construct the proper energy

expression for excited singlet states originating from a closed shell ground state. Considering first a spin-‐conserving transition from a close shell ground state and assume without loss of generality that the transition takes place in the α-manifold. In that case we can write the occupied excited state KS-orbitals generated from the unitary transformation of (1) as27

!i' = cos["# i ]!io$ + sin["# i ]!iv$ ; i = 1,occ / 2

(8)

and

!i' = !i ; i = occ / 2 + 1,occ ,

(9)

whereas the corresponding KS-determinant is given by

! M = "1'"2' ..."i'" 'j ..."n' .

(10)

Here ! M represents a mixed spin-state42 that is half singlet and half triplet. Further,

! i (i = 1,occ / 2) is a set of eigenvalues to (V!! )+ U!! W!! = 1" , (11) where ! is a diagonal matrix of dimension occ/2 whereas U!! is the part of the U matrix that runs over the occupied {!i ;i = 1,occ / 2} and virtual {!a ;i = 1,vir / 2} ground state orbitals of α-spin.27, 49 Further !io" =

occ/2

# (W

""

) ji ! j (12)

j

and


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vir/2

! = $ (V "" )ai# a (13) v" i

a

Finally ! is determined in such a way that

#

occ/2 i=1

sin 2 [!" i ] = 1 , corresponding to the

constraint that exactly one electron charge is involved in the transition. The orbitals defined in (12) and (13) have been referred to as Natural Transition Orbitals (NTO)50 since they give a more compact description of the excitations than the canonical orbitals. Thus, a transition that involves several i ! a replacements among canonical orbitals can often be described by a single replacement !io" # !iv" in terms of NTO’s. We note again that the set in (8) is obtained from the unitary transformation (1) among the occupied

{!i ;i = 1,occ / 2} and virtual {!a ;i = 1,vir / 2} ground state orbitals of α-spin with U represented by U!! . For a spin-flip transition from a closed shell ground state the unitary transformation (1) among the occupied ground state orbitals {!i ;i = 1,occ / 2} of α-spin and the virtual ground state orbitals {!a ;i = vir / 2 + 1,vir} of β-spin yields the occupied excited state orbitals

!i'' = cos["# i' ]!io$ + sin["# i' ]!i % ; i = 1,occ / 2 v

(14)

and

!i'' = !i ; i = occ / 2 + 1,occ ,

(15)

whereas the corresponding KS-determinant is given by

! T = "1'"2' ..."i'" 'j ..."n' .

(16)

Here ! T represents a triplet state. Further, ! i' (i = 1,occ / 2) are the eigenvalues to

(V !" )+ U !" W !" = 1# ' (17)


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where U !" is the part of U that runs over the virtual ground state orbitals of β-spin and occupied ground state orbitals of α-spin. Finally !

o" i

=

occ/2

$ (W

#"

) ji ! j (18)

j=1

and a=vir

!i " = v

%

(V "# )ai$ a . (19)

a=vir/2+1

With the energy of ! T given by ET and that of ! M as EM , we can write the singlet transition energy as42

!ES = 2EM " ET " E0 , (20) provided that U!! = U "! . This implies that V!! = V "! ; W !! = W "! and ! = ! ' . Further

!io" of (8) and (14) become identical as do the spatial parts of !iv" and !i " in the same two v

equations. In CV(∞)-DFT use is made of the U !" matrix obtained from a CV(2)-TD singlet calculation. Straightforward manipulations27 allow us finally to write down the singlet transition energy in a compact and closed form as !ES =

occ/2

$ sin ["# 2

i=1

+

i

](% i v& ' % i o& )

1 occ/2 occ/2 2 sin ["# j ]sin 2 ["# i ](K i o& i o& j o& j o& + K i v& i v& j v& j v& ' 2K i v& i v& j o& j o& + K i v( i v( j o& j o& ] (21) $ $ 2 i=1 j=1

occ/2 occ/2

$ $ sin["# i=1

j=1

j

]cos["# j ]sin["# i ]cos["# i ](2K i o& i v& j o& j v& ' K i o& i v( j o& j v( )

Here the indices i o! , j o! ,i v! and j v! refer to α-spin orbitals with the spatial parts

!io" ,! oj " ,!iv" and ! vj " , respectively. Likewise the indices i ! , j ! , i o


o

v!

and j ! refer to βv

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spin orbitals with the spatial parts !i " ,! j " ,!i " and ! j " , respectively. o

o

v

v

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In deriving

(21)

we have applied the Tamm-‐Dancoff approximation28 by neglecting integrals of the type K i o! i v! j v! j o! in the third term on the right hand side of (21) as well as numerically very small contributions that are third order in sin[!" i ] . These approximations are introduced to ensure that (21) for excitations that are described by several displacements converge to the CV(2)-‐TD expression for a singlet transition, as we shall discuss shortly. The expression for the corresponding triplet transition energy reads !ET =

occ/2

$ sin ["# 2

i=1

i

](% i v& ' % i o& )

1 occ/2 occ/2 2 + $ $ sin ["# j ]sin 2 ["# i ](K i o& i o& j o& j o& + K i v( i v( j v( j v( ' K i v( i v( j o& j o& ] (22) 2 i=1 j=1 occ/2 occ/2

$ $ sin["# i=1

j=1

j

]cos["# j ]sin["# i ]cos["# i ]K i o& i v( j o& j v(

We shall from now on refer to CV(∞)-DFT calculations in which (21) and (22) are used as CV(∞)-TD. Please note that for single displacement transitions where !" j = # / 2 for j=i and !" j = 0 for j ! i , we get the same energy whether the TD-approximation is used or not since only the first and second term in (22) contributes to !ES .

2.2 Computational Details All calculations have been performed with the developer’s version of the Amsterdam Density Functional (ADF) package. The Slater-‐type basis supplied by ADF30 version 2010 was of triple-‐ζ quality31 with an additional set of polarized functions (TZP). From extensive TDDFT studies23-‐25 on dyes, we estimate that our excitation energies


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are within 0.1 eV to 0.2 eV of those obtained by a converged basis set. Use was made of the local density approximation LDA(VWN)32 in the parameterization due to Vosko, Wilk, and Nusair. For the calculation of the exact Hartree-‐Fock exchange integrals over Slater-‐type orbitals, a double fitting procedure has been employed.33 All ground state structures were adopted from the work of Richard and Herbert16. 3 Results and Discussion We present in Table 1 the calculated 1 A1g ! 1B2u and 1 A1g ! 1B3u excitation energies !E( 1B2u ) and !E( 1B3u ) ,respectively, for linear acenes employing the TDDFT and CV(2)-‐TD methods in conjunction with the LDA(VWN) functional. Also shown are the calculated oscillatory strengths of the two excitations. Extensive studies16 have shown that the 1 A1g ! 1B2u transition essentially can be represented by the HOMO → LUMO promotion whereas 1 A1g ! 1B3u is a nearly equal mixture of the HOMO-‐1 → LUMO and HOMO → LUMO+1 promotions. Excitation energies ( !ES ) due to transitions from a closed shell ground state to a singlet excited state have in CV(2)-‐TD the simple form27 vir/2 occ/2

vir/2 occ/2

vir/2 occ/2

!ES = # # U aiU ai (! a " !i ) + 2 # # U aiU bj K ai,bj " # # U ai U bj K ai ,bj . (23) a

i

a,b

i, j

a,b

i, j

Here superscript “bars” refer to orbitals of ! ! spin whereas i, a without bars represent ! ! orbitals. The corresponding energies in TDDFT has a somewhat more complex form4 but depend again on (! a ! !i ) , K ai,bj , K ai ,bj and U .


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It follows from Table 1 and Figure 1a that the experimental energy gap !E = !E( 1B2u ) ! "E( 1B3u ) starts out positive at naphthalene (nr=2) with Δ! =

0.53 !" before turning negative at anthracene (nr=3) where Δ! = −0.04 !". For larger linear acenes Δ! becomes increasingly negative reaching Δ! = −0.85 !" at hexacene. Thus, experimentally, Δ!( !!!! ) is seen to drop faster in energy than Δ!( !!!! ). The ordering of the calculated CV(2)-‐TD excitation energies based on LDA(VWN) is correct for naphthalene as well as for all other linear acenes, however, the gap differs from the experimental Δ! by almost 0.4 eV for naphthalene, anthracene and hexacene. This difference is smaller for napthacene and pentacene, Table 1 and Figure 1b. It can be seen that the main contribution to this deviation comes from the underestimation of Δ!( !!!! ). Thus, the root mean square deviation (RMSD) value for Δ!( !!!! ) is 0.49 eV, while it is only 0.13 eV for Δ!( !!!! ). As for the TDDFT results based on LDA(VWN), the deviation from the experimental gaps is larger in absolute terms and the calculated Δ! has the wrong sign for naphthalene, Table 1 and Figure 1c. It happens, because the Δ!( !!!! ) values for TDDFT are lower than those for CV(2)-‐TD while the Δ!( !!!! ) estimates for TDDFT are higher than for CV(2)-‐TD. As a result, the RMSD value for Δ!( !!!! ) increases for TDDFT by approximately 0.2 eV and reaches 0.71 eV. On the other hand, on average the TDDFT estimate of Δ!( !!!! ) is as accurate as for CV(2)-‐TD. Thus, the TDDFT Δ!( !!!! ) values for napthacene, pentacene and hexacene are closer to experiment than those due to CV(2)-‐TD while the opposite is true for


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naphthalene and anthracene. The RMSD value for TDDFT is 0.14 eV compare to 0.13 eV for CV(2)-‐TD, Table 1.

It follows from the discussion given above that neither CV(2)-‐TD nor TDDFT

are able to give a quantitative description of Δ! as a function of nr with LDA, in line with previous TDDFT studies2,3,8,9 using both pure density functionals and hybrids. The source of the error is in all cases primarily Δ!( !!!! ) that is too low compared to experiment. However, Δ!( !!!! ) is in addition seen to be slightly too high. We have in order to analyze our findings further displayed the orbital energy dependent part Δ! =

!"# !

!

!"" !

!

!!" !!" (!! − !! ) (24)

in the expression for Δ!! according to CV(2)-‐TD , eq 23, along with the K-‐integral dependent part Δ!! − Δ! in Table 2 and Figures 2a-‐b as a function of nr for LDA in the case of Δ!( !!!! ) and Δ!( !!!! ). We see that both the slope and the absolute value of Δ!( !!!! ) and Δ!( !!!! ) are determined by Δ! and that the contributions to Δ!! from the K-‐integrals amount to 0.3 eV for Δ!( !!!! ) and 0.1 eV for Δ!

!

!!! ,where both contributions diminish slightly with nr. Turning now to the CV(∞)-‐DFT scheme we note that the excitation energy

(21) for a transition that involves a single promotion ! → ! such as 1 A1g ! 1B2u has the simple form27 Δ!!!"

!

!

!

! → ! = !! − !! + ! !!!!! + ! !!!!! − 2!!!"" + !!!!! (25).

Here i is the HOMO and a the LUMO. For CV(2) we obtain for the same transition according to eq 23


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Δ!!!"

!

Page 14 of 29

! → ! = !! − !! + 2!!"!" − !!!!! (26).

For HF these two expressions are identical27 since !!!!! = !!!!! = 0 and !!"!" = −!!!"" . However for any of the popular functionals this is not the case. Thus, the two expressions will give rise to different excitation energies for the same functional, as we shall see. For transitions consisting of two or more promotions such as 1

!" !

A1g ! 1B3u , the expression of eq (21) for Δ!!

contains contributions that are

second and fourth order in sin. The contributions from the latter terms diminish rapidly in magnitude with the number of promotions due to the constraint

#

occ/2 i=1

sin 2 [!" i ] = 1 . Thus, for four equally participating promotions with

sin 2 [!" i ] = 1 / 4 (i=1,4), the fourth order prefactors sin 4 [!" i ] are small (1/16) and the terms in (21) to second order in sin will dominate since sin 2 [!" i ] = 1 / 4 (i=1,4) . In the limit of many more participating promotions (21) will converge to (23).

It follows from Table 3 and Figure 1d that the CV(∞)-‐DFT results using LDA

are in excellent agreement with experiment for both Δ!( !!!! ) and Δ!( !!!! ) throughout the range of linear acenes. The RMSD for Δ!( !!!! ) is 0.06 eV whereas that for Δ!( !!!! ) is 0.13 eV. Thus, CV(∞)-‐DFT clearly represents an improvement over TDDFT and CV(2)-‐TD for LDA. The improvement is as anticipated most noticeable for Δ!( !!!! ) where the RMSD was 0.71 eV for TDDFT and 0.49 eV for CV(2)-‐TD. For Δ!( !!!! ) all three methods have a similar RMSD.

The magnitude of 1/2!!!!! + 1/2!!!!! − 2!!!"" + !!!!! from eq 25 is given in

Table 4 and Figure 2c for !!!! as Δ!( !!!! )-‐ Δ!

!

!!! . It is on the average 0.8 eV and

is seen to decrease slightly with n. The magnitude of this term depends strongly on the


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spatial extents of a and i. If the densities of a and i are strongly overlapping the first two terms might be canceled by the last two. This has been found for some π → π∗ excitation.27 At the other extreme is charge transfer transitions35-‐38 where there is no overlap and where the centers of the two densities are separated by a large distance R. In that case the two last terms reduce to 1/R. Many studies have shown that charge transfer transitions are better described by (25) than by (26).35-‐39 In the case at hand the 0.8 eV supplied by 1/2!!!!! + 1/2!!!!! − 2!!!"" + !!!!! is just enough to bring the CV(∞)-DFT values calculated for Δ!( !!!! ) in line with experiment whereas the corresponding CV(2)-TD contribution given by 2!!"!" − !!!!! is too small (0.30 eV). Given that the excitation energy for one-orbital transitions in CV(∞)-DFT is expressed by (10) rather than (11), we can in general expect it to differ from that calculated by CV(2)-TD or TDDFT irrespective of whether or not it exhibits charge transfer character. Also shown in Table 4 and Figure 2d is Δ!( !!!! )-‐ Δ!

!

!!! calculated by

CV(∞)-DFT for the linear acenes. It is much smaller than Δ!( !!!! )-‐ Δ! more in line with the corresponding Δ!( !!!! )-‐ Δ!

!

!

!!!

and

!!! values obtained by CV(2)-TD

or TDDFT, Table 2 and Figure 2b. This is consistent with that 1Lb (1B3u) is described by more than one orbital substitution and the fact that the energy expression (21) for CV(∞)-TD in the limit of excitations with several orbital replacements affords CV(2)-TD transition energies given by (23). Kuritz19 et al. have provided an interesting analysis of how LRC functionals improve the description of the 1La transitions.


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We have extended our benchmark calculations also to include the 15 nonlinear acenes shown in Figure 3. The resulting singlet transition energies involving 1La and 1Lb are shown in Table 5 for TDDFT, CV(2)-TD and CV(∞)-TD based on LDA. For 1La CV(∞)-TD with a RMSD of 0.24 eV is seen to perform better than

CV(2)-TD

(RMSD=0.40 eV) and especially TDDFT ( 0.52 eV). In fact our results are of a similar quality to the best results obtained by long-‐range corrected (LRC) functionals.16 We note again that the 1La transitions involves a single HOMO ! LUMO orbital displacement with !" i = # / 2 . For 1Lb all three methods performs equally well with RMSD’s of 0.16 eV (TDDFT), 0.15 eV(CV(2)-‐TD) and 0.19 eV (CV(∞)-‐TD), respectively. It is interesting to note that the LRC-‐functionals16 in this case performs much poorer with RMSD’s around 0.4 eV. Thus , CV(∞)-‐TD at the simple LDA level is the only scheme of the methods discussed here that gives a balanced description of π → π∗ transitions involving a single orbital displacement (1La) as well as a π → π∗ transitions with more than one displacement.

3. Concluding Remarks

We have here tested the all order constricted variational density functional

theory CV(∞)-‐TD in conjunction with LDA as a possible inexpensive method for the computational screening of organic dyes. Our initial test set was made up of linear poly-‐acenes ranging from naphthalene with two to hexacene with six rings. For these systems we studied the energy difference Δ! between the excitation energies of the first two π → π∗ transitions 1La (1B2u) and 1Lb (1B3u) as a function of the


16

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number of rings nr. These systems are known to be challenging for DFT based electronic structure methods.

Our study revealed that CV(∞)-‐TD combined with LDA gave as accurate a

description of Δ! as the sophisticated DFT/MRCI11-‐12 and TDDFT/LRC

13-‐16

approaches for linear acenes while representing a clear improvement over TDDFT2,3,8,9 using pure functionals or hybrids. This is interesting given the low cost and simplicity of the LDA functional and the computational efficiency of CV(∞)-‐TD which is a factor two more expensive than TDDFT or CV(2)-‐TD schemes alone.

It is very well known that transitions involving a single substitution such as

1L (1B ) a 2u

often are better described by (25) that originates from the ∆!"# 39-‐46

energy expression than from (26) due to TDDFT-‐TD . This point has recently been demonstrated also to be the case for π → π∗ transitions27,47 in conjugated hydrocarbons. Also Casida48 has suggested to use (25) in place of (26) for single substitution transitions involving charge transfer as have others.35-‐37 Thus, as such our CV(∞)-‐TD method does not bring anything new.

The novel aspect of our scheme is that it provides a seamless

transition from a single substitution excitation where the ∆!"# type energy expression (25) is required to multi-‐replacement transitions where (26) is appropriate. The all order constricted variation theory represents in addition a sound theoretical framework that contains both ∆!"# and adiabatic TDDFT-‐TD as special cases. We should finally note that that the CV(∞)-‐TD scheme is a simple form of more refined theories such as SCF-‐CV(∞)-‐DFT27 in which U is optimized with


17


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Page 18 of 29

respect to E[ ! (n)+, ! (n)! ] rather than E[ ! (2)+, ! (2)! ] and R-‐CV(∞)-‐DFT or RSCF-‐CV(∞)-‐ DFT in which we also introduce relaxation of the orbitals not participating directly in the excitation.29 However, in spite of its simplicity the CV(∞)-‐TD scheme was shown recently to afford an average deviation from experiment of approximately 0.3 eV for both semi-‐local and hybrid functionals in an extensive study of π → π∗ transitions27,47 involving conjugated hydrocarbons. The current study on linear acenes adds further to the notion that CV(∞)-‐TD might be a useful and cost efficient supplement to TDDFT in the study of dyes. We have in addition provided an extensive theoretical analysis of how CV(∞)-‐TD complement TDDFT. Acknowledgement. This work was supported by NSERC. The computational resources of WESTGRID were used for all calculations. T.Z. thanks the Canadian Government for a Canada Research Chair and the Humboldt foundation for a generous award.


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References (1) Platt, J. R. J. Chem. Phys. 1949, 17, 484–496. (2) Grimme, S.; Parac, M. Chem. Phys. Chem. 2003, 4, 292–295. (3) Parac, M.; Grimme, S. Chem. Phys. 2003, 292, 11–21. (4) Casida, M. E. In Recent Advances in Density Functional Methods; D. P. Chong, Ed.; World Scientific: Singapore, 1995; Vol. 1, pp. 155-192. (5) Gross, E. K. U.; Dobson, J. F.; Petersilka, M. Top. Curr. Chem. 1996, 181, 81–172. (6) Bauernschmitt, R.; Ahlrichs, R. Chem. Phys. Lett. 1996, 256, 454–464. (7) Time-Dependent Density Functional Theory, Lecture Notes Physics 706; Marques, M. A. L., Ulrich, C. A., Nagueira, F., Rubio, A., Burke, K., Gross, E. K. U., Eds.; Springer: Berlin-Heidelberg, 2006. (8) Jacquemin, D.; Wathelet, V.; Perpete, E. A.; Adamo, C. J. Chem. Theory Comput. 2009, 5, 2420–2435. (9) Goerigk, L.; Grimme, S. J. Chem. Phys. 2010, 132, 184103. (10) Dierksen, M.; Grimme, S. J. Phys. Chem. A 2004, 108, 10225-10237. (11) Grimme, S.; Waletzke, M. J. Chem. Phys. 1999, 111, 5645–5655. (12) Marian, C.; Gilka, N. J. Chem. Theory Comput. 2008, 4, 1501–1515. (13) Jacquemin, D.; Perpete, E. A.; Scuseria, G. E.; Ciofni, I.; Adamo, C. J. Chem. Theory Comput. 2008, 4, 123–135. (14) Jacquemin, D.; Perpete, E. A.; Ciofni, I.; Adamo, C. Theor. Chem. Acc. 2011, 128, 127–136. (15) Wong, B. M.; Hsieh, T. H. J. Chem. Theory Comput. 2010, 6, 3704–3712. (16) Richard, R. M.; Herbert, J. M. J. Chem. Theory Comput. 2011, 7, 1296–1306. (17) Stein, T.; Kronik, L.; Baer, R. J. Am. Chem. Soc. 2009, 131, 2818–2820. (18) Stein, T.; Kronik, L.; Baer, R. J. Chem. Phys. 2009, 131, 244119. (19) Kuritz, N.; Stein, T.; Kronik, R. B. L. J. Chem. Theory Comput. 2011, 7, 2408–2415. (20) Grimme, S.; Neese, F. J. Chem. Phys. 2007, 127, 154116. (21) Grimme, S. J. Chem. Phys. 2006, 124, 034108. (22) Karton, A.; Tarnopolsky, A.; Lamere, J. F.; Schatz, G. C.; Martin, J. M. L. J. Phys. Chem. A 2008, 112, 12868–12886. (23) Goerigk, L.; Moellmann, J.; Grimme, S. Phys. Chem. Chem. Phys. 2009, 11, 4611– 4620. (24) Goerigk, L.; Grimme, S. J. Phys. Chem. A 2009, 113, 767–776. (25) Goerigk, L.; Grimme, S. J.Chem.Theory Comput. 2011,7,3272-3277. (26) Head-Gordon, M.; Rico, R. J.; Oumi, M.; Lee, T. J. Chem. Phys. Lett. 1994, 219, 21–29. (27) Ziegler,T.; Krykunov, M. and Cullen, J. J. Chem. Phys. 2012, 136, 124107. (28) Hirata, S; Head-‐Gordon M. Chem. Phys. Lett. 1999, 314, 291-‐299. (29) Ziegler, T; Krykunov, M. J. Chem. Phys. 2010, 133, 074104. (30) te Velde, G.; Bickelhaupt, F. M.; van Gisbergen, S. J. A.; Fonseca Guerra, C.; Baerends, E. J.; Snijders, J. G.; Ziegler, T. J. Comput. Chem. 2001, 22, 931-‐967. (31) Van Lenthe, E.; Baerends, E. J. J. Comput. Chem. 2003, 24, 1142-‐1156. (32) Vosko, S. H.; Wilk, L.; Nusair, M. Can. J. Phys. 1980, 58, 1200-‐1211.


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(33) Krykunov, M. ; Ziegler, T. and van Lenthe, E. J. Phys. Chem. A, 2009, 113, 11495–11500 (34) Biermann,B.; Schmidt,W. J.Am.Chem.Soc. 1980,102,3163-‐3173 (35) Ziegler, T.; Seth, M.; Krykunov, M.; Autschbach, J.; Wang, F. (Theochem) J.Mol. Struct. 2009, 914, 106–109. (36) Ziegler, T.; Seth, M.; Krykunov, M.; Autschbach, J. J. Chem. Phys. 2008, 129, 184114. (37) Ziegler, T.; Seth, M.; Krykunov, M.; Autschbach, J.; Wang, F. J.Chem. Phys. 2009, 130, 154102. (38) Cullen,J.; Krykunov,M. and Ziegler,T. Chem.Phys. 2011,391,11-‐18. (39) Ziegler,T.; Rauk, A. Baerends, E.J. Chem. Phys., 1976, 16, 209-‐217. (40) Slater, J.C. and Wood,J.H. Int.J. Quantum Chem. 1971, Suppl. 4 , 3-‐34. (41) Slater,J.C. Adv. Quantum.Chem. 1972, 6 ,1-‐31. (42) Ziegler,T.; Rauk, A. and Baerends, E.J. Theor.Chim. Acta 1977, 43, 261-‐271. (43 ) Levy,M. and Perdew,J.P. Phys.Rev. 1985, A 32, 2010-‐2021. (44) Nagy,A. Phys. Rev. A 1996, 53, 3660-‐3663. (45) Besley,N; Gilbert,A and Gill, P. J. Chem. Phys. 2009,130, 124308. (46) Gavnholt, J. ; Olsen, T.; Engelund, M; Schiotz,J. Phys. Rev. B 2008, 78, 075441. (47) Kowalczyk,T.; Yost,S.R.; Van Voorhis, T. J.Chem.Phys. 2011, 134,054128. (48) Cassida, M.E. ; Gutierrez, F. ; Guan, J. Gadea, F.-‐X. ; Salahub, D. and Daudey, J.-‐P. J. Chem. Phys. 2000,113, 7062-‐7071. (49) Amos, A.T; Hall, G.G. Proc. Roy.Soc. 1961, A263 , 483-‐493. (50) Martin,R.L. J.Chem.Phys. 2003, 118 , 4775-‐4777.


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Table 1. LDA(VWN) based 1La(1B2u) and 1Lb(1B3u) singlet excitation energies (in eV) for acenes with CV(2)-TD and TDDFT c b Experiment CV(2)-TDd f TDDFT

1B 2u

1B

a

a 1B 1B nr ∆E ∆E ∆E 3u 2u 3u 2 4.66 4.13 -0.53 4.27 4.21 0.06 0.0545 0.0000 4.07 4.41 0.34 3 3.60 3.64 -0.04 3.13 3.60 -0.47 0.0578 0.0003 2.91 3.78 -0.87 4 2.88 3.39 -0.51 2.38 3.21 -0.83 0.0533 0.0013 2.15 3.35 -1.20 5 2.37 3.12 -0.75 1.86 2.94 -1.08 0.0469 0.0033 1.62 3.08 -1.46 6 2.02 2.87 -0.85 1.47 2.74 -1.27 0.0399 0.0071 1.22 2.86 -1.64 RMSD 0.49 0.13 0.71 0.14 a ∆E = ∆E(1B2u) − ∆E(1B3u). bOscillator strength.cRef. 34.dTamm-Dancoff approximation. eNumber of rings e

1B 2u

1B 3u

a

1B 2u


1B 3

21


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Page 22 of 29

Table 2. Contributions from orbital energies and K integrals to the calculated excitation energies (eV) for CV(2)-‐TDd and LDA. Cal. 1B2u Cal. 1B3u a ! cn ∆! Δ!! − Δϵ r ∆! (Δ!! − Δϵ) 2 3.95 0.32 4.14 0.07 3 2.83 0.30 3.55 0.05 4 2.11 0.27 3.16 0.05 5 1.60 0.26 2.90 0.04 6 1.23 0.24 2.71 0.03 aOrbital energy contributions to the excitation energies, see eqs 25 and 26. bContributions from K-‐integrals to the excitation energies. cNumber of rings. d Tamm-Dancoff approximation.


22

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Table 3. CV(∞)-TD singlet excitation energies (in eV) for linear acenes with LDA. Number bExperiment LDA a 1 1 of rings B2u B3u ∆E 1B2u 1B3u a∆E 2 4.66 4.13 0.53 4.73 4.39 0.34 3 3.60 3.64 -0.04 3.68 3.73 -0.05 4 2.88 3.39 -0.51 2.91 3.32 -0.41 5 2.37 3.12 -0.75 2.35 3.03 -0.68 6 2.02 2.87 -0.85 1.93 2.82 -0.89 RMSD 0.06 0.13 a ∆E = ∆E(1B2u) − ∆E(1B3u). bRef. 34


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Page 24 of 29

Table 4. Contributions from orbital energies and K integrals to the calculated excitation energies (eV) using CV(∞)-‐TD and LDA. Cal. 1B2u Cal. 1B3u a ! cn ∆! Δ!! − Δϵ r ∆! (Δ!! − Δϵ) 2 3.95 0.78 4.14 0.25 3 2.83 0.85 3.55 0.18 4 2.11 0.80 3.16 0.16 5 1.60 0.75 2.90 0.13 6 1.23 0.70 2.71 0.11 aOrbital energy contributions to the excitation energies, see eqs 25 and 26. bContributions from K-‐integrals to the excitation energies. cNumber of rings.


24

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Table 5. LDA(VWN) based 1La(1B2u) and 1Lb(1B3u) singlet excitation energies (in eV) for nonlinear acenes with TDDFT , CV(2)-TD and CV(∞)-TD 1 1 La Lb Moleculea

TDDFT CV(2)b CV(∞)

b

Expc TDDFT CV(2)b CV(∞)b Exp.c

1

3.88

3.99

4.09

4.24

3.59

3.61

3.74

3.76

2

3.39

3.60

3.69

3.71

3.46

3.47

3.58

3.53

3

4.16

4.27

4.41

4.54

3.74

3.76

4.00

3.89

4

3.37

3.51

3.83

3.89

3.43

3.41

3.58

3.43

5

2.96

3.16

3.98

3.63

3.10

3.07

3.68

3.22

6

2.53

2.78

2.90

2.85

3.22

3.23

3.37

--

7

3.36

3.51

3.61

3.73

3.22

3.22

3.33

3.37

8

2.84

3.02

3.56

3.22

3.07

3.09

3.43

3.06

9

3.29

3.35

3.55

3.80

3.14

3.14

3.26

3.30

10

3.07

3.18

3.37

3.84

2.98

2.99

3.08

3.32

11

3.08

3.18

4.12

3.84

3.20

3.21

3.31

3.31

12

3.11

3.21

3.47

3.86

3.01

3.03

3.17

3.14

13

2.72

2.84

3.34

3.40

3.02

3.02

3.22

3.23

14

2.55

2.80

3.02

2.86

2.99

2.99

3.08

--

15

3.69

3.90

3.87

3.72

3.85

3.86

3.94

--

RMSD

0.52

0.40

0.24

0.16

0.15

0.19

a

The structures are shown in Fig. 1. Tamm-Dancoff approximation. c The experimental values are taken from Ref. 2. b


25


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Figure 1. Excitation energies for the 1La and 1Lb transitions in linear acenes as a function of the number of rings according to (a)experiment, (b) CV(2)-TD, (c) TDDFT and (d) CV(∞)-TD. 153x107mm (150 x 150 DPI)


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Figure 2. Contributions from orbital energies and K integrals to the calculated excitation energies (eV) in linear acenes for CV(2)-TD with respect to (a) 1B2u (1La) and (b) 1B3u (1Lb) as well as CV(∞)-TD with respect to (c) 1B2u (1La) and (d) 1B3u (1Lb) 150x107mm (150 x 150 DPI)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 3. Structures for nonlinear acenes 183x238mm (150 x 150 DPI)


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graphical abstract 215x279mm (150 x 150 DPI)


Accurate Theoretical Description of the 1La and 1Lb Excited States in

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